Quantity

Remark: Something's Limit as Quality

§ 393

In something, its limit as quality is essentially its determinateness. If, however, by limit we mean quantitative limit, then when, for example, a field alters its limit it still remains what it was before, a field. If on the other hand its qualitative limit is altered, then since this is the determinateness which makes it a field, it becomes a meadow, wood, and so on. A red, whether brighter or paler, is still red; but if it altered its quality it would cease to be red, would become blue or some other colour. The determination of magnitude as quantum reached above, namely that it has a permanent substratum of being which is indifferent to its determinateness, can be found in any other example.

§ 394

By magnitude quantum is meant, as in the examples cited, not quantity; and it is chiefly for this reason that this foreign term must be used.

The definition of magnitude given in mathematics likewise concerns quantum. A magnitude is usually defined as that which can be increased or diminished. But to increase means to make the magnitude more, to decrease, to make the magnitude less. In this there lies a difference of magnitude as such from itself and magnitude would thus be that of which the magnitude can be altered. The definition thus proves itself to be inept in so far as the same term is used in it which was to have been defined. Since the same term must not be used in the definition, the more and less can be resolved into an affirmative addition which, in accordance with the nature of quantum, is likewise external, and a subtraction, as an equally external negation. It is this external form both of reality and of negation which in general characterises the nature of alteration in quantum. In that imperfect expression, therefore, one cannot fail to recognise the main point involved, namely the indifference of the alteration, so that the alteration's own more and less, its indifference to itself, lies in its very Notion.

Pure quantity has not as yet any limit or is not as yet quantum; and even in so far as it becomes quantum it is not bounded by limit but, on the contrary, consists precisely in not being bounded by limit, in having being-for-self within it as a sublated moment. The presence in it of discreteness as a moment can be expressed by saying that quantity is simply the omnipresent real possibility within itself of the one, but conversely that the one is no less absolutely continuous.

§ 400

In thinking that is not based on the Notion, continuity easily becomes mere composition, that is, an external relation of the ones to one another, in which the one is maintained in its absolute brittleness and exclusiveness. But it has been shown that the one essentially and spontaneously (an und für sich selbst) passes over into attraction, into its ideality, and that consequently continuity is not external to it but belongs to it and is grounded in its very nature. It is just this externality of continuity for the ones to which atomism clings and which ordinary thinking finds it difficult to forsake. Mathematics, on the other hand, rejects a metaphysics which would make time consist of points of time; space in general — or in the first place the line — consist of points of space; the plane, of lines; and total space of planes. It allows no validity to such discontinuous ones. Even though, for instance, in determining the magnitude of a plane, it represents it as the sum of infinitely many lines, this discreteness counts only as a momentary representation, and the sublation of the discreteness is already implied in the infinite plurality of the lines, since the space which they are supposed to constitute is after all bounded.

§ 401

It is the notion of pure quantity as opposed to the mere image of it that Spinoza, for whom it had especial importance, has in mind when he speaks of quantity as follows:

'Quantity is conceived by us in two manners, to wit, abstractly and superficially, as an offspring of imagination or as a substance, which is done by the intellect alone. If, then, we look at quantity as it is in the imagination, which we often and very easily do, it will be found to be finite, divisible, and composed of parts; but if we look at it as it is in the intellect and conceive it, in so far as it is a substance, which is done with great difficulty, then as we have already sufficiently shown, it will be found to be infinite, without like, and indivisible. This, to all who know how to distinguish between the imagination and the intellect, will be quite clear.'

§ 402

More specific examples of pure quantity, if they are wanted, are space and time, also matter as such, light, and so forth, and the ego itself: only by quantity, as already remarked, is not to be understood quantum. Space, time and the rest, are expansions, pluralities which are a coming-out-of-self, a flowing which, however, does not pass over into its opposite, into quality or the one, but as a coming-out-of-self they are a perennial self-production of their unity.

§ 403

Space is this absolute self-externality which equally is absolutely uninterrupted, a perpetual becoming-other which is self-identical; time is an absolute coming-out-of-itself, a generating of the one, (a point of time, the now) and immediately the annihilation of it, and again the continuous annihilation of this passing away; so that this spontaneous generating of non-being is equally a simple self-sameness and self-identity. As regards matter as quantity, among the seven surviving propositions of the first dissertation of Leibniz there is one, the second, which runs: Non omnino improbabile est, materiam et quantitatem esse realiter idem. In fact, the distinction between these two concepts is simply this, that quantity is a determination of pure thought, whereas matter is the same determination in outer existence. The determination of pure quantity belongs also to the ego which is an absolute becoming-other, an infinite removal or all-round repulsion to the negative freedom of being-for-self which, however, remains utterly simple continuity — the continuity of universality or being-with-self uninterrupted by the infinitely manifold limits, by the content of sensations, intuitions, and so forth. Those who reject the idea of plurality as a simple unity and besides the Notion of it, to wit, that each of the many is the same as every other, namely, a one of the many — since here we are not speaking of the many as further determined, as green, red, and so on, but of the many considered in and for itself — demand also a representation of this unity, will find plenty of instances in those continua which exhibit the deduced Notion of quantity as present in simple intuition.

Remark 2: The Kantian Antinomy of the Indivisibility and the Infinite Divisibility of Time, Space and Matter

§ 404

It is the nature of quantity, this simple unity of discreteness and continuity, that gives rise to the conflict or antinomy of the infinite divisibility of space, time, matter, etc.

§ 405

This antinomy consists solely in the fact that discreteness must be asserted just as much as continuity. The one-sided assertion of discreteness gives infinite or absolute dividedness, hence an indivisible, for principle: the one-sided assertion of continuity, on the other hand, gives infinite divisibility.

§ 406

It is well known that the Kantian Critique of Pure Reason sets up four (cosmological) antinomies, the second of which deals with the antithesis constituted by the moments of quantity.

§ 407

Kantian antinomies will always remain an important part of the critical philosophy; they, more than anything else, brought about the downfall of previous metaphysics and can be regarded as a main transition into more recent philosophy since they, in particular, helped to produce the conviction of the nullity of the categories of finitude in regard to their content which is a more correct method than the formal one of a subjective idealism, according to which their defect is supposed to be, not what they are in themselves, but only that they are subjective. But this exposition with all its merits is imperfect; its course is impeded and tangled, and also it is false in regard to its result, which presupposes that cognition has no other forms of thought than finite categories. In both respects these antinomies deserve a more exact critical appraisal which will not only throw more light on their standpoint and method but will also free the main point at issue from the useless form into which it has been forced.

§ 408

In the first place, I remark that Kant wanted to give his four cosmological antinomies a show of completeness by the principle of classification which he took from his schema of the categories. But profounder insight into the antinomial, or more truly into the dialectical nature of reason demonstrates any Notion whatever to be a unity of opposed moments to which, therefore, the form of antinomial assertions could be given. Becoming, determinate being, etc., and any other Notion, could thus provide its particular antinomy, and thus as many antinomies could be constructed as there are Notions.

Ancient scepticism did not spare itself the pains of demonstrating this contradiction or antimony in every notion which confronted it in the sciences.®

§ 409

Further, Kant did not take up the antinomy in the Notions themselves, but in the already concrete form of cosmological determinations. In order to possess the antinomy in its purity and to deal with it in its simple Notion, the determinations of thought must not be taken in their application to and entanglement in the general idea of the world, of space, time, matter, etc; this concrete material must be omitted from consideration of these determinations which it is powerless to influence and which must be considered purely on their own account since they alone constitute the essence and the ground of the antinomies.

§ 410

Kant's conception of the antinomies is that they are 'not sophisms but contradictions which reason must necessarily come up against' (a Kantian expression); and this is an important view. 'Reason, when it sees into the ground of the natural illusion of the antinomies is, it is true, no longer imposed on by them but yet continues to be deceived.' The Kantian solution, namely, through the so-called transcendental ideality of the world of perception, has no other result than to make the so-called conflict into something subjective, in which of course it remains still the same illusion, that is, is as unresolved, as before. Its genuine solution can only be this: two opposed determinations which belong necessarily to one and the same Notion cannot be valid each on its own in its one-sidedness; on the contrary, they are true only as sublated, only in the unity of their Notion.

§ 411

The Kantian antinomies on closer inspection contain nothing more than the quite simple categorical assertion of each of the two opposed moments of a determination, each being taken on its own in isolation from the other. But at the same time this simple categorical, or strictly speaking assertoric statement is wrapped up in a false, twisted scaffolding of reasoning which is intended to produce a semblance of proof and to conceal and disguise the merely assertoric character of the statement, as closer consideration will show.

§ 412

The relevant antinomy here concerns the so-called infinite divisibility of matter and rests on the antithesis of the moments of continuity and discreteness which are contained in the Notion of quantity.

The thesis of the same as expounded by Kant runs thus:

'Every composite substance. in the world consists of simple parts, and nowhere does there exist anything but the simple or what is compounded from it.'

§ 413

To the simple, the atom, there is here opposed the composite, which is a very inferior determination compared to the continuous. The substrate given to these abstractions, namely, substances in the world, here means nothing more than things as sensuously perceived and it has no influence on the antinomy itself; space or time could equally well be taken. Now since the thesis speaks only of composition instead of continuity it is really as it stands an analytical or tautological proposition. That the composite is not in its own self a one, but only something externally put together and consisting of what is other than itself, this is its immediate determination. But the other of the composite is the simple. It is therefore tautological to say that the composite consists of the simple. To ask of what something consists is to ask for an indication of something else, the compounding of which constitutes the said something. If ink is said to consist simply of ink, the meaning of the inquiry after the something else of which it consists has been missed and the question is not answered but only repeated. A further question then is whether that which is under discussion is supposed to consist of something or not. But the composite is simply that which is supposed to be a combination of something else. If, however, the simple which is the other of the composite is taken only as relatively simple and is itself composite, too, then the question still remains unanswered. What ordinary thinking has in mind is, perhaps, only some composite or other of which something or other, too, would be assigned as its simple, such particular something being composite on its own account. But what is under discussion here is the composite as such.

§ 414

Now as regards the Kantian proof of the thesis this, like all the Kantian proofs of the rest of the antinomial propositions, makes the detour of being apagogic, a detour which will prove to be quite superfluous.

'Assume', he begins, 'that composite substances do not consist of simple parts; then if all composition were thought away no composite part and (since we have just assumed that there are no simple parts) no simple part — hence nothing at all — would remain; consequently, no substance would have been given.'

This conclusion is quite correct: if nothing but composite substances exist and all that is composite is thought away, then nothing whatever remains; this will be conceded, but this tautological redundance could be omitted and the proof straightway begin with what follows, namely:

'Either it is impossible to think away all composition, or else there remains after such removal in thought something which is not composite, that is, the simple.

'In the first case, however, the composite again would not consist of substances (because with these, composition is only a contingent relation of substances [In addition to the redundance of the proof itself there is here also a redundance of language 'because with these' (namely the substances) 'composition is only a contingent relation of substances'.] which, apart from such relation, must still persist on their own account). Now since this case contradicts what was assumed, only the second case is left: namely, that all composite substances consist of simple parts.'

§ 415

The very reason which is the main point, and in face of which all that precedes is completely superfluous, is mentioned by the way, in a parenthesis. The dilemma is this: either the composite persists, or else the simple. If the former, that is, the composite, persists, then what persists would not be substances, for composition is for these only a contingent relation; but substances do persist, therefore, what persists is the simple.

§ 416

It is clear that the apagogical detour could be omitted and the thesis:

'composite substance consists of simple parts', could be directly followed by the reason: because composition is merely a contingent relation of substances, and is therefore external to them and does not concern the substances themselves. If the composition is in fact contingent then, of course, substances are essentially simple. But this contingency which is the sole point at issue is not proved but straightway assumed, and casually, too, in a parenthesis-as something self-evident or of secondary importance. True, it goes without saying that composition is a contingent and external determination; but if the point at issue were only a contingent togetherness instead of continuity, it would not be worth while constructing an antinomy about it — or rather it would not be possible to formulate one. Therefore, the assertion that the parts are simple is, as remarked, only a tautology.

§ 417

The apagogical detour thus contains the very assertion which should result from it. The proof therefore can be put more concisely thus:

Let us assume that substances do not consist of simple parts but are only composite. But now all composition can be thought away (for it is only a contingent relation); after its removal, therefore, there are no substances left unless we assume that they consist of simple parts. But substances we must have for we have assumed them; everything is not meant to vanish, something must be left over, for we have presupposed something persistent which we called substance. Therefore this something must be simple.

§ 418

To complete the whole, we have still to consider the conclusion which runs as follows:

'From this it follows, as a direct consequence, that all things in the world without exception are simple entities, that composition is only an external state of them, and that reason must think the elementary substances as simple entities.'

Here we see the externality, that is contingency, of composition put forward as a consequence after it had already been introduced parenthetically and used in the proof.

§ 419

Kant strongly protests that he is not looking for sophisms in the conflicting statements of the antinomy for the purpose, as it were, of special pleading. But the defect of the proof in question is not so much that it is a sophism, as that its laboured, tortuous complexity serves no other purpose than to produce the merely outward semblance of a proof and partially to obscure the quite transparent fact that what was supposed to emerge as a consequence is, parenthetically, that on which the proof hinges; that there is no proof at all, but only an assumption.

The antithesis runs:

'No composite thing in the world consists of simple parts and nowhere in the world does there exist anything simple.'

The proof has equally an apagogical turn and, in a different way, is just as faulty as the previous one.

'Suppose', it runs, 'that a composite thing as a substance consists of simple parts. Because all external relation, and consequently all composition of substances, is possible only in space, therefore, the space occupied by the composite substance must consist of as many parts as those of which the composite substance consists. Now space does not consist of simple parts but of spaces. Therefore each part of the composite substance must occupy a space.'

'But the absolutely primary parts of everything composite are simple.'

'Therefore the simple occupies a space.'

'Now since every real thing that occupies a space comprises a manifold of mutually external parts and is consequently composite, consisting of substances, it would follow that the simple is a composite substance-which is self-contradictory.'

This proof can be called a whole nest (to use an expression elsewhere employed by Kant) of faulty procedure.

§ 420

In the first place, its apagogical form is a groundless illusion. For the assumption that whatever is substantial is spatial, but that space does not consist of simple parts is a direct assertion which is made the immediate ground of what is to be proved, and with this there is an end to the proving of the antithesis.

§ 421

Next, this apagogical proof begins with the proposition: 'that all composition from substances is an external relation', but oddly enough immediately forgets it. For it then goes on to conclude that composition is possible only in space, that space does not consist of simple parts, and that therefore the real thing occupying a space is composite. But once composition is assumed as an external relation, then spatiality itself (in which alone composition is supposed to be possible) is for that very reason an external relation for the substances, which does not concern them or affect their nature any more than anything else does that can be inferred from the determination of spatiality. For this very reason, the substances ought not to have been put into space.

§ 422

Further, it is assumed that the space in which the substances here are placed does not consist of simple parts, because it is an intuition, that is, according to the Kantian definition, a representation which can only be given through a single object, and is not a so-called discursive concept. This Kantian distinction between intuition and concept has, as everyone knows, given rise to a deal of nonsense about the former, and to avoid the labour of comprehension the value and sphere of intuition have been extended to the whole field of cognition. What is pertinent here is just this, that space, and also intuition itself, must be grasped in terms of their Notions if, that is, we want really to comprehend. And thus the question would arise whether space, even though a simple continuity for intuition, ought not to be grasped, in accordance with its Notion, as consisting of simple parts, or whether it would be involved in the same antinomy which applied only to substance. As a matter of fact, if the antinomy is grasped abstractly, it concerns, as we remarked, quantity as such, and hence equally space and time.

§ 423

But it is assumed in the proof that space does not consist of simple parts; this therefore ought to have been the reason for not placing the simple in this element which is incompatible with the nature of the simple. There is also involved here a clash between the continuity of space and composition; the two are confused with each other, the former being substituted for the latter (which results in a quaternio terminorum in the conclusion). With Kant, space has the express determination of being 'sole and single, its parts resting only on limitations, so that they do not precede the one, all-embracing space as, so to speak, its component parts from which it could be compounded.' Here continuity is quite correctly and definitely predicated of space in denial of its composition from parts. On the other hand, in the argument the placing of substances in space is supposed to involve 'a manifold of mutually external parts' and, more particularly, 'consequently a composite'. Yet, as we have quoted, the way in which manifoldness is present in space is expressly intended to exclude composition and component parts antecedent to the unity of space.

§ 424

In the remark to the proof of the antithesis we are also expressly reminded of the other fundamental conception of the critical philosophy, namely, that we have a notion of bodies only as appearances or phenomena; as such, however, they necessarily presuppose space as the condition of the possibility of all outer appearance. If by substances we are meant here to understand only bodies as we see, touch, taste them, and so on, then we are not really discussing them as they are in their Notion but only as sensuously perceived. The proof of the antithesis, then, amounted in short to this: all our visual, tactile, and other experience shows us only what is composite; even the best microscopes and the keenest knives have not enabled us to come across anything simple. Then neither should reason expect to come across anything simple.

§ 425

When we look more closely into the opposition of this thesis and antithesis, freeing their proofs from all pointless redundancy and tortuousness, we see that the proof of the antithesis dogmatically assumes continuity (by placing substances in space) and also that the proof of the thesis, by assuming that composition is the mode of relation of substances, dogmatically assumes the contingency of this relation, and hence assumes that substances are absolute ones. Thus the whole antinomy reduces to the separation of the two moments of quantity and the direct assertion of them as absolutely separate. When substance, matter, space, time, etc., are taken only as discrete, they are absolutely divided; their principle is the one. When they are taken as continuous, this one is only a sublated one; division remains a divisibility, it remains the possibility of division, as a possibility, without actually reaching the atom. Now even if we stop at the determination given in what has been said about these antitheses, then the moment of the atom is contained in continuity itself, for this is simply the possibility of division; just as said dividedness, discreteness, sublates all distinction of the ones — for each of the simple ones is what the other is-consequently, also contains their sameness and hence their continuity.

Since each of the two opposed sides contains its other within itself and neither can be thought without the other, it follows that neither of these determinations, taken alone, has truth; this belongs only to their unity. This is the true dialectical consideration of them and also the true result.

§ 426

Infinitely more ingenious and profound than this Kantian antinomy are the dialectical examples of the ancient Eleatic school, especially those concerning motion, which likewise are based on the Notion of quantity and in it find their solution. To consider them here, too, would be too lengthy a business; they concern the Notions of space and time and can be dealt with at the same time as these subjects and in the history of philosophy. They reflect the greatest credit on the intelligence of their inventors they have for result the pure being of Parmenides, in that in them is demonstrated the dissolution of all determinate being; they are thus in themselves the flux of Heraclitus. For this reason they deserve a more thorough consideration than the usual explanation that they are just sophisms; which assertion sticks to empirical perception, following the procedure of Diogenes (a procedure which is so illuminating to ordinary common sense) who, when a dialectician pointed out the contradiction contained in motion, made no effort to reason it out but, by silently walking up and down, is supposed to have referred to the evidence of sight for an answer. Such assertion and refutation is certainly easier to make than to engage in thinking and to hold fast and resolve by thought alone the complexities originating in thought, and not in abstruse thought either, but in the thoughts spontaneously arising in ordinary consciousness.

§ 427

The solutions propounded by Aristotle of these dialectical forms merit high praise, and are contained in his genuinely speculative Notions of space, time and motion. To infinite divisibility (which, being imagined as actually carried out, is the same as infinite dividedness, as the atoms) on which is based the most famous of those proofs, he opposes continuity, which applies equally well to time as to space, so that the infinite, that is, abstract plurality is contained only in principle [an sich], as a possibility, in continuity. What is actual in contrast to abstract plurality as also to abstract continuity, is their concrete forms, space and time themselves, just as these latter are abstract relatively to matter and motion. What is abstract has only an implicit or potential being; it only is as a moment of something real. Bayle, who finds Aristotle's solution of the Zenonic dialectic 'pitoyable', does not understand the meaning of the statement that matter is only potentially infinitely divisible; he rejoins that if matter is infinitely divisible, then it actually contains an infinite number of parts, that, therefore, this infinite is not an infinite en putssance but an infinite that really and actually exists. On the contrary, divisibility itself even is only a possibility, not an existing of the parts, and the plurality as such is posited in the continuity only as a moment, as sublated. Acute understanding, in which Aristotle, too, is certainly unsurpassed, is not competent to grasp and to decide on speculative Notions, any more than the crudity of sensuous conception instanced above is adequate to refute the reasoning of Zeno. Such intellect commits the error of holding such mental fictions, such abstractions, as an infinite number of parts, to be something true and actual; but this sensuous consciousness does not let itself be brought beyond the empirical element to thought.

§ 428

The Kantian solution of the antinomy likewise consists solely in the supposition that reason should not soar beyond sensuous perception and should take the world of appearance, the phenomenal world, as it is. This solution leaves the content of the antinomy itself on one side; it does not attain to the nature of the Notion of its determinations, each of which, isolated on its own, is null and is in its own self only the transition into its other, the unity of both being quantity, in which they have their truth.

B. CONTINUOUS AND DISCRETE MAGNITUDE

§ 429

1. Quantity contains the two moments of continuity and discreteness. It is to be posited in both of them as determinations of itself. It is already their immediate unity, that is, quantity is posited at first only in one of its determinations, continuity, and as such is continuous magnitude.

§ 430

Or we may say that continuity is indeed one of the moments of quantity which requires the other moment, discreteness, to complete it. But quantity is a concrete unity only in so far as it is the unity of distinct moments. These, are, therefore, also to be taken as distinct, but are not to be resolved again into attraction and repulsion, but are to be taken as they are in their truth, each remaining in its unity with the other, that is, remaining the whole. Continuity is only coherent, compact unity as unity of the discrete; posited as such it is no longer only a moment but the whole of quantity, continuous magnitude.

§ 431

2. Immediate quantity is continuous magnitude. But quantity is not an immediate at all; immediacy is a determinateness the sublatedness of which is quantity itself. It is, therefore, to be posited in the determinateness immanent in it, and this is the one. Quantity is discrete magnitude.

Discreteness is, like continuity, a moment of quantity but it is itself also the whole of quantity just because it is a moment in it, in the whole, and therefore as a distinct moment it does not stand outside the whole, outside its unity with the other moment. Quantity is in itself asunderness, and continuous magnitude is this asunderness continuing itself without negation as an internally self-same connectedness. But discrete magnitude is this asunderness as discontinuous, as interrupted. With this plurality of ones, however, we are not again in the presence of the plurality of atoms and the void, repulsion in general. Because discrete magnitude is quantity, its discreteness is itself continuous. This continuity in the discrete consists in the ones being the same as one another, or in having the same unity. Discrete magnitude is, therefore, the asunderness of the manifold one as self-same, not the manifold one in general but posited as the many of a unity.